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Geometry

# 3D Coordinate Geometry: Level 2 Challenges

Point $$P$$ is some point on the surface of the sphere ${(x-1)}^{2}+{(y+2)}^{2}+{(z-3)}^{2}=1 .$ What is the shortest possible distance between $$P$$ and $$O=(0, 0, 0)?$$

If the two lines \begin{align} \frac {x-1 }{k } &=\frac { y+1 }{2 } =z,\\ \\ \\ \frac {x+2 }{-3 } & =1-y =\frac{z+2}{k} \end{align} are perpendicular to each other, then what is the value of $$k?$$

The points $$A$$, $$B$$, $$P$$, and $$Q$$ all lie on one line, with $$A=(-3, 5, 8).$$

Point $$P$$ lies in the $$xy$$-plane between $$A$$ and $$B$$ such that the distance from $$A$$ to $$P$$ is twice the distance from $$P$$ to $$B$$. Furthermore, $$Q$$ sits on the $$z$$-axis such that the distance from $$A$$ to $$Q$$ is twice the distance from $$Q$$ to $$B$$.

If $$B=(x,y,z),$$ what is the value of $$x+y+z?$$

An infinite column is centered along the $$z$$-axis. It has a square cross section of side length equal to 10. It is cut by the plane $$4x - 7y + 4z = 25.$$. What is the area of the surface cut?

If the point $$Q(a, b, c)$$ is the reflection of the point $$P(-6, 2, 3)$$ about the plane $$3x-4y+5z-9=0$$. Determine the value of $$a+b+c$$.

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